1549055384-Symplectic_Geometry_and_Topology__Eliashberg_

LECTURE 5. VARIATIONAL INTEGRATORS 405

issues discussed in lectures 3 and 4 are examples. Other examples are the direct

variational approach to questions in black hole dynamics given by Wald [1993] and

the development of variational asymptotics (see Holm [1996], Holm, Marsden and

Ratiu [1998b], and references therein). In such studies, it is the variational principle

that is t he center of attention.

One can derive in a natural way the entire differential geometric structures,

including momentum mappings, directly from the variational approach. This de-

velopment begins by removing the endpoint condition 8q(a) = 8q(b) = 0. Equa-

tion (5.19) becomes

l

b. ( 8£ d 8L) 8L · 1b

d6(q(-)). 8q(·) = a 8q' [)qi - dt [)qi dt + [)qi 8q' a' (5.21)

but now the left side operates on more general 8q and the last term on the right side

need not vanish. That last term of (5.21) is a linear pairing of the function 8L/8qi,

a function of qi and qi, with the tangent vector oqi. Thus, one may consider it

a 1-form on TQ; namely the 1-form (8L/8qi)dqi. This is exactly the Lagrange

1-form, and we can summarize this as follows:

Theorem 5.5. Given a Ck Lagrangian L, k;::: 2, there exists a unique ck-^2 map-

ping DeLL: Q--+ T*Q, defined on the second order submanifold

Q = { ~:; (0) E T

2

Q I q is a C^2 curve in Q}

of T^2 Q , and a unique Ck-l 1-form GL on TQ, such that, for all C^2 variations

q€ ( t)'

l

d6(q(-)) ·8q(·) = a b DeLL (ddt2q) (dq) lb

2 ·8qdt+ GL dt ·o'q a' (5.22)

where

8q(t) , = -d I -d I qe(t).
de e=O dt t =O
The 1-form so defined is called the Lagrange 1-form.
Indeed, uniqueness and local existence are consequences of the calcula-
tion (5.19), and the coordinate independence of the action, and then global ex-
istence follows.
Thus, using the variational principle, the Lagrange 1-form e L is the "boundary
part" of the the functional derivative of the action when the boundary is varied. The
analogue of the symplectic form is the (negative of) exterior derivative of 8 L; i.e.,

[),L =: -d8£.

Lagrangian flows are symplectic. One of Lagrange's basic discoveries was that

the solutions of the Euler-Lagrange equations give rise to a symplectic map. It is a

curious twist of history that he did this without the machinery of either differential
forms, of the Hamiltonian formalism or of Hamilton's principle itself. (See Marsden
and Ratiu [1998] for an account of this history.)
Assuming that L is regular, the variational principle gives as we have seen, co-
ordinate independent second order ordinary differential equations. We temporarily
denote t he vector field on TQ so obtained by X , and its flow by Ft. Consider the
restriction of 6 to the subspace CL of solutions of the variational principle. The